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V. Christianto, http://www.sciprint.org, email: admin@sciprint.org F. Smarandache, Dept. of Mathematics, Univ. of New Mexico, Gallup, USA 1. Introduction It is known that quaternion number has wide application in theoretical physics and engineering fields alike, in particular to describe Maxwell electrodynamics. In the meantime, recently this quaternion number has also been used to draw fractal graph. The present note is intended as an introduction to this very interesting study, i.e. to find linkage between quaternion/biquaternion number, quantum mechanical equation (Schrödinger equation) and fractal graph. Hopefully this note will be found useful for subsequent study. 2. An alternative derivation of Schrödinger-type equation

In this section we will make an attempt to re-derive a Schrödinger-type equation, but with a new

definition of total energy. In this regard, it seems worth noting here that it is more proper to use Noether’s expression of total

energy in lieu of standard derivation of Schrödinger’s equation ( mpE 2/2�= ). According to Noether’s theorem [4], the total energy of the system corresponding to the time translation invariance is given by:

( ) 2

0

222 .4.).2/( ckdrrcwmcE µπγ =+= �∞

(1)

where k is dimensionless function. It could be shown, that for low-energy state the total energy could be far

less than 2mcE = . In this regard, interestingly Bakhoum [5] has also argued in favor of using 2mvE = for expression of total energy, which expression could be traced back to Leibniz. Therefore it

seems possible to argue that expression 2mvE = is more generalized than the standard expression of special relativity, in particular because the total energy now depends on actual velocity [4].

We start with Bakhoum’s assertion 2mvE = , instead of more convenient form 2

smcE = . This notion would imply [5]:

222222 ... vcmcpH o−= . (2) Therefore, for phonon speed in the limit p�0, we write [6]:

pcpE s.)( ≡ . (3)

In the first approximation, we could derive Klein-Gordon-type relativistic equation from equation (2), as follows. By introducing a new parameter:

)/( cvi=ζ , (4) then we can rewrite equation (2) in the known procedure of Klein-Gordon equation:

422222 .. cmcpE oζ+= , (5)

where 2mvE = . [5] By using known substitution: tiE ∂∂= /.� , ip /∇= � , (6)

and dividing by ( )2c� , we get Klein-Gordon-type relativistic equation:

Ψ=Ψ∇+∂Ψ∂− − ./.2'22

oktc , (7)

where

2

�/' cmk oo ζ= . (8) One could derive Dirac-type equation using similar method. Nonetheless, the use of new parameter (4)

seems to be indirect solution, albeit it simplifies the solution, because here we can use the same solution from Klein-Gordon equation.

Alternatively, one could derive a new quantum relativistic equation, by noting that expression of total

energy 2mvE = is already relativistic equation. We will derive here an alternative approach using Ulrych’s [7] method to get relativistic wave function from this expression of total energy [4].

vpmvE .2 == (9) Taking square of this expression, we get:

222 .vpE = (10) or

222 / vEp = (11) Now we use Ulrych’s substitution [7]:

( ) ( )[ ] 2pqAPqAP =−− µµ , (12)

and introducing standard substitution in Quantum Mechanics (6), one gets:

( ) ( )[ ] Ψ∂∂=Ψ−− − 22 )/..( tivqAPqAP �µ

µ , (13)

or

( )( )[ ] 0)/./( 2 =Ψ∂∂−−∇−−∇− tviqAiqAi ���µµ

µµ . (14)

This equation is comparable to Schrödinger equation for a charged particle interacting with an external electromagnetic field [8]:

( )( )[ ] [ ]Ψ+∂∂−=Ψ−∇−−∇− )(2/.2 xmUtmiqAiqAi µµµµ �� . (15)

In other words, we could re-derive Schrödinger-type equation for a charged particle from Ulrych’s approach [7].

Alternatively, one can use similar assertion as Schrödinger described in his original equation:

mpmvE /22 == (16) Using the same method (equation 12), we get:

( )( )[ ] 0)/..( =Ψ∂∂−−∇−−∇− timqAiqAi ���µµ

µµ . (17)

For m�1, one recovers standard Schrödinger equation [8]. 3. Introduction to Quaternion number Let us begin with a few definitions of numbers. It is known that complex numbers are an extension to the real numbers. They can be seen as two dimensional vectors where also multiplication is defined. We define it as z=a+bi, where a real part, b imaginary part, and i2= -1. Complex: z=t+xi Quaternion: z = t + xi + yj + zk Octonion: z = t + xi + yj + zk+ aE + bI + cJ + wK Where: z=(t,V), t=scalar, V=vector. Furthermore, we can define conjugate complex of z: biaz −=�

which has properties

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222 zbazz =+=�

For application of these numbers in quantum physics, see [7][9][10]. 4. Introduction to Biquaternion number Biquaternion numbers are an extension of quaternion to four dimensions [11]. They can be seen as four dimensional vectors (with one scalar and a vector in three space). In physics they are also used in relativity; it is also very useful to describe Maxwell electrodynamics in its original form [7][12]. We could define:

z = a + bi + cj + dk where i2=j2=-1 and k2=jk=1

For those not familiar with the matrices of Biquaternion and quaternion algebra, here are the tables:

Biquaternion math table

i j k i -1 k -j j k -1 -i k -j -i 1

Quaternion math table

i j k i -1 k -j j -k -1 i k j -i -1

In both quaternion and Biquaternion math i^2 = -1. The Biquaternion rules provide for one real variable, two complex variables (i and j) and one variable which Charles Muses refers to as countercomplex (k). In quaternion math there is one real variable and three complex variables. In Biquaternion math, unlike quaternion math, the commutative law holds, that is reversing the order of multiplication doesn't change the product.

One other concept that mathematicians like to dwell on is the idea of a "ring". There is one ring in quaternion and Biquaternion math, "ijk". If you start anywhere in this ring and proceed to multiply three variables in a loop, backwards or forwards, you get the same number, 1 for Biquaternion, -1 for quaternion [13].

From this viewpoint, we can find further extension of Schrödinger type equation described above to biquaternion form [7][10].

Now we’re ready to find simulation of this number via fractal graph. [11]

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5. Fractal graph (examples) A few examples of fractal graph from quaternion number can be found at www.fraktalstudio.de, and www.bugman123.com. The following graphs were drawn with Dofo-Zon (www.mysticfractal.com), and FractalExplorer ().

Picture 1. Random quaternion (Dofo-Zon)

Picture 2. Random quaternion (Dofo-Zon)

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Picture 3. Random quaternion (Dofo-Zon)

Picture 4. Random quaternion (Dofo-Zon)

Picture 5. Random quaternion (Dofo-Zon)

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Picture 6. Quaternion, Q*Q*Q+Q’ (Fractal Explorer)

Picture 6. Quaternion, Q*Q*Q*Q+Q’ (Fractal Explorer)

Concluding remarks We have explored some of those stunning images created using the notion of quaternion numbers to draw fractal graphs. This application of quaternion numbers in physics are known, therefore it could be expected that such quaternion/biquaternion fractal graphs will also be found useful in theoretical physics alike. This will be the subject of further exploration. Dec. 14th, 2005

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References [1] Lam, T.Y., D. Leep, J-P. Tignol, “Biquaternionic algebras and quartic extensions,” Publications mathematiques de l’I.H.E.S. tome 77 (1993) 63-102. URL http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1993__77_/PMIHES_1993__77__63_0/PMIHES_1993__77__63_0.pdf [2] Childress, S., & J. Wrkich, “Introduction to biquaternion algebras,” http://math.fullerton.edu/aagnew/urs/ursmeetings.html [3] Majernik, V., “Evolution of the universe in the presence of quaternionic field,” Fac. Rer. Nat. (2000), Physica (39) 9-24. URL: http://publib.upol.cz/~obd/fulltext/Physica%2039/Physica%2039_01.pdf [4] Ying-Qiu, G., “Some properties of the spinor soliton,” Advances in Applied Clifford Algebras 8 No. 1 (1998) 17-29. [5] Bakhoum, E., “Fundamental disagreement of wave mechanics with relativity,” arXiv:physics/0206061 (2002) p. 5-7. [5a] Bakhoum, E., physics/0207107. [5b] Bakhoum, E., email communication, Sept. 24th 2005. [6] Consoli, M., arXiv:gr-qc/0204106 (2002). [6a] Consoli, M., & E. Costanzo, arXiv:hep-ph/0311317 (2003). [7] Ulrych, S, arXiv:hep-th/9904170 (1999); [7a] Ulrych, S., physics/0009079. [8] Kiehn, R.M., “An interpretation of wavefunction as a measure of vorticity,” http://www22.pair.com/csdc/pdf/cologne.pdf (1989). [9] Castro., C., & M. Pavsic, “The extended relativity in Clifford-spaces,” commissioned review to appear in Int. J. Mod. Phys. D (2004). [10] Kravchenko, V., & V.V. Kravchenko, arXiv:math-ph/0305046 (2003). [11] www.fraktalstudio.de [12] Gauguin, P.A., “Unified Dirac-Maxwell as spacetime portal,” http://www.europeanufosurvey.com/en/en.stargate.pdf [13] www.mysticfractal.com; [13a] www.ultrafractal.com [14] http://www.student.math.uwaterloo.ca/~pmat370/FractalRefs.html

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Fractal links Some useful links for drawing fractal from quaternion numbers:

Name Description URL link QuaSZ QuaSZ Mac

Primarily for exploring quaternions, hypernions, octonions, cubics, and complexified quats.

www.mysticfractal.com

Hydra This program graphs 3-D slices of formulas based on 4-D complex number planes, currently supporting quaternion, hypernion, and user-customized quad types of the Mandelbrot set and Julia sets.

www.mysticfractal.com

Fractal Agent Fractal Commander

Freeware programs orignally written to draw escape-type fractals using every conceivable complex math function. And now convergent and orbit-trap types, and extended basic complex math to hypercomplex and quaternion math

http://www.geocities.com/SoHo/Lofts/5601

Quaternion Julia Set VRML Server

A CGI engine used to generate VRML quaternion Julia sets.

http://www.ecs.wsu.edu/~hart

Source: http://home.att.net/~Paul.N.Lee/Fractal_Software.html